Coarse-grained (CG) models reduce the number of degrees of freedom in a system, allowing the dynamics of large systems to be studied for longer times. Many CG models have been developed since the way of generating a CG system highly depends on the question one is interested in. It is common to treat one or multiple molecules as one CG bead by their center of mass. However, this way fails to study non-bonded particles since the motion of free particles disperses them and disintegrates the cluster. The conservative potentials, arising from CG mapping schemes[1] for non-bonded atomistic particles, are studied. These schemes map atomistic particles to fluid element-like cells whose centers lie on a regular, cubic lattice. Equilibrium atomistic molecular dynamics trajectories for a liquid and gaseous Lennard-Jones fluid are converted to CG ones, from which CG probability distribution functions are calculated. We extend the past work[2] by applying fuzzy switching functions at the boundary regions of subcells. Atomistic particles are shared by multiple subcells and produce a continuous change to CG mass elements when they cross the boundary of subcells. A full mass matrix is required to describe the behaviour of the CG potential. As the boundary gets fuzzier, we observe a transition from discrete to continuous distributions for diagonal mass elements. This transition produces a quantitative measure to when continuum theories like fluctuating hydrodynamics are appropriate to describe the system. Non-zero correlations among all CG variables are calculated, and found to depend strongly on the degree of fuzziness. In particular, those for the diagonal mass elements decrease in magnitude and there exists a specific value of the fuzziness for which the correlations are zero. Other correlations are strong only when the fuzziness is quite large, so there is a tradeoff between the complexity of the interactions in the system and the degree of fuzziness between the subcells. However, if the number of particles in a subcell is large enough, and the fuzziness is moderate, the CG potential is found to be well-approximated by a generalized, quadratic function. For a homogeneous system, a few unique parameters are needed to reconstruct the CG potential. These potential parameters can also be evaluated from theory using atomistic particle distribution functions. [1] Lynn, H.; Thachuk, M. Equations of motion for position-dependent coarse-grain mappings obtained with Mori-Zwanzig theory. J. Chem. Phys. 2019, 150, 024108. [2] Luo, S.; Thachuk, M. Conservative Potentials for a Lattice-Mapped Coarse-Grained Scheme. J. Phys. Chem. A 2021, 125, 6486-6497.